Hypersurfaces In Projective Space Research Articles

Quantum cohomology from mixed Higgs-Coulomb phases

Abstract We generalize Coulomb-branch-based gauged linear sigma model (GLSM)–computations of quantum cohomology rings of Fano spaces. Typically such computations have focused on GLSMs without superpotential, for which the low energy limit of the GLSM is a pure Coulomb branch, and quantum cohomology is determined by the critical locus of a twisted one-loop effective superpotential. We extend these results to cases for which the low energy limit of the GLSM includes both Coulomb and Higgs branches, where the latter is a Landau-Ginzburg orbifold. We describe the state spaces and products of corresponding operators in detail, comparing a geometric phase description, where the operator product ring is quantum cohomology, to the description in terms of Coulomb and Higgs branch states. As a concrete test of our methods, we compare to existing mathematics results for quantum cohomology rings of hypersurfaces in projective spaces.

Integral points on algebraic subvarieties of period domains: from number fields to finitely generated fields

We show that for a variety which admits a quasi-finite period map, finiteness (resp. non-Zariski-density) of S-integral points implies finiteness (resp. non-Zariski-density) of points over all mathbb {Z}-finitely generated integral domains of characteristic zero. Our proofs rely on foundational results in Hodge theory due to Deligne, Griffiths, and Schmid, and Bakker-Brunebarbe-Tsimerman. We give straightforward applications to arithmetic locally symmetric varieties, the moduli space of smooth hypersurfaces in projective space, and the moduli of smooth divisors in an abelian variety.

Variational Hodge conjecture for complete intersections on hypersurfaces in projective space

In this paper we give a new and simplified proof of the variational Hodge conjecture for complete intersection cycles on a hypersurface in projective space.

On genus one mirror symmetry in higher dimensions and the BCOV conjectures

Abstract The mathematical physicists Bershadsky–Cecotti–Ooguri–Vafa (BCOV) proposed, in a seminal article from 1994, a conjecture extending genus zero mirror symmetry to higher genera. With a view towards a refined formulation of the Grothendieck–Riemann–Roch theorem, we offer a mathematical description of the BCOV conjecture at genus one. As an application of the arithmetic Riemann–Roch theorem of Gillet–Soulé and our previous results on the BCOV invariant, we establish this conjecture for Calabi–Yau hypersurfaces in projective spaces. Our contribution takes place on the B-side, and together with the work of Zinger on the A-side, it provides the first complete examples of the mirror symmetry program in higher dimensions. The case of quintic threefolds was studied by Fang–Lu–Yoshikawa. Our approach also lends itself to arithmetic considerations of the BCOV invariant, and we study a Chowla–Selberg type theorem expressing it in terms of special $\Gamma $ -values for certain Calabi–Yau manifolds with complex multiplication.

An arithmetic enrichment of Bézout’s Theorem

The classical version of B\'ezout's Theorem gives an integer-valued count of the intersection points of hypersurfaces in projective space over an algebraically closed field. Using work of Kass and Wickelgren, we prove a version of B\'ezout's Theorem over any perfect field by giving a bilinear form-valued count of the intersection points of hypersurfaces in projective space. Over non-algebraically closed fields, this enriched B\'ezout's Theorem imposes a relation on the gradients of the hypersurfaces at their intersection points. As corollaries, we obtain arithmetic-geometric versions of B\'ezout's Theorem over the reals, rationals, and finite fields of odd characteristic.

Modular forms from Noether–Lefschetz theory

We enumerate smooth rational curves on very general Weierstrass fibrations over hypersurfaces in projective space. The generating functions for these numbers lie in the ring of classical modular forms. The method of proof uses topological intersection products on a period stack and the cohomological theta correspondence of Kudla and Millson for special cycles on a locally symmetric space of orthogonal type. The results here apply only in base degree 1, but heuristics for higher base degree match predictions from the topological string partition function.

Low-degree rational curves on hypersurfaces in projective spaces and their fan degenerations

We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves with balanced normal bundle, and reprove some results on irreducibility of spaces of rational curves of low degree.

Diophantine problems and p-adic period mappings

We give an alternative proof of Faltings’s theorem (Mordell’s conjecture): a curve of genus at least two over a number field has finitely many rational points. Our argument utilizes the set-up of Faltings’s original proof, but is in spirit closer to the methods of Chabauty and Kim: we replace the use of abelian varieties by a more detailed analysis of the variation of p-adic Galois representations in a family of algebraic varieties. The key inputs into this analysis are the comparison theorems of p-adic Hodge theory, and explicit topological computations of monodromy. By the same methods we show that, in sufficiently large dimension and degree, the set of hypersurfaces in projective space, with good reduction away from a fixed set of primes, is contained in a proper Zariski-closed subset of the moduli space of all hypersurfaces. This uses in an essential way the Ax–Schanuel property for period mappings, recently established by Bakker and Tsimerman.

Hypergeometric decomposition of symmetric K3 quartic pencils

We study the hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard–Fuchs differential equations; we count points using Gauss sums and rewrite this in terms of finite-field hypergeometric sums; then we match up each differential equation to a factor of the zeta function, and we write this in terms of global L-functions. This computation gives a complete, explicit description of the motives for these pencils in terms of hypergeometric motives.

The Chow Ring of a Cubic Hypersurface

Abstract We study the product structure on the Chow ring (with rational coefficients) of a cubic hypersurface in projective space and prove that the image of the product map is as small as possible.

Derived categories of BHK mirrors

We prove a derived analogue to the results of Borisov, Clarke, Kelly, and Shoemaker on the birationality of Berglund-Hübsch-Krawitz mirrors. Heavily bootstrapping off work of Seidel and Sheridan, we obtain Homological Mirror Symmetry for Berglund-Hübsch-Krawitz mirror pencils to hypersurfaces in projective space.

Topologies of Nodal Sets of Random Band‐Limited Functions

AbstractIt is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band‐limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular, the results apply to random monochromatic waves and to random real algebraic hypersurfaces in projective space. © 2018 Wiley Periodicals, Inc.

Remarks on higher-rank ACM bundles on hypersurfaces

In terms of the number of generators, one of the simplest non-split rank-3 arithmetically Cohen–Macaulay bundles on a smooth hypersurface in P5 is 6-generated. We prove that a general hypersurface in P5 of degree d≥3 does not support such a bundle. We also prove that a smooth positive dimensional hypersurface in projective space of even degree does not support an Ulrich bundle of odd rank and determinant of the form OX(c) for some integer c. This verifies some cases of conjectures we discuss here.

Structure of the Main Tensor of Conformally Connected Torsion Free Space. Conformal Connections on Hypersurfaces of Projective Space

We give a definition of a conformally connected space with an angular metric of an arbitrary signature. We present basic formulas and classes of such spaces. We obtain the decomposition of the main tensor of a conformally connected torsion free space into irreducible gauge invariant summands and prove that all affine connections obtained from the Levi–Civita connection via the normalization transformation have the same Weyl conformal tensor. We describe all conformal torsion free connections on hypersurfaces of a projective space and give some examples. We construct a global conformal connection on a hyperquadric of the projective space.

Quantum geometry of resurgent perturbative/nonperturbative relations

For a wide variety of quantum potentials, including the textbook ‘instanton’ examples of the periodic cosine and symmetric double-well potentials, the perturbative data coming from fluctuations about the vacuum saddle encodes all non-perturbative data in all higher non-perturbative sectors. Here we unify these examples in geometric terms, arguing that the all-orders quantum action determines the all-orders quantum dual action for quantum spectral problems associated with a classical genus one elliptic curve. Furthermore, for a special class of genus one potentials this relation is particularly simple: this class includes the cubic oscillator, symmetric double-well, symmetric degenerate triple-well, and periodic cosine potential. These are related to the Chebyshev potentials, which are in turn related to certain mathcal{N} = 2 supersymmetric quantum field theories, to mirror maps for hypersurfaces in projective spaces, and also to topological c = 3 Landau-Ginzburg models and ‘special geometry’. These systems inherit a natural modular structure corresponding to Ramanujan’s theory of elliptic functions in alternative bases, which is especially important for the quantization. Insights from supersymmetric quantum field theory suggest similar structures for more complicated potentials, corresponding to higher genus. Our approach is very elementary, using basic classical geometry combined with all-orders WKB.

Derived Knörrer periodicity and Orlov’s theorem for gauged Landau–Ginzburg models

We prove a Knörrer-periodicity-type equivalence between derived factorization categories of gauged Landau–Ginzburg models, which is an analogy of a theorem proved by Shipman and Isik independently. As an application, we obtain a gauged Landau–Ginzburg version of Orlov’s theorem describing a relationship between categories of graded matrix factorizations and derived categories of hypersurfaces in projective spaces, by combining the above Knörrer periodicity type equivalence and the theory of variations of geometric invariant theory quotients due to Ballard, Favero and Katzarkov.

A note on Cartan's second main theorem for holomorphic curve intersecting hypersurface

In this paper, we establish a second main theorem for holomorphic curve intersecting hypersurfaces in projective space with level of truncation and give some applications. Furthermore, we show that the defect relation in Ru's result [7] may be improved for some special divisors.

Compact Invariant Sets for Global Holomorhic Foliations

We study compact invariant sets for holomorphic foliations on Stein manifold. As application, we show some dynamical properties concerning minimal sets (with singularities) of foliations and real analytic Levi-flat hypersurfaces in projective spaces.

On the Fukaya category of a Fano hypersurface in projective space

This paper is about the Fukaya category of a Fano hypersurface $X \subset \mathbb{CP}^n$. Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify considerably. The first part of the paper is devoted to establishing the main structures of the Fukaya category in the monotone case: the closed-open string maps, weak proper Calabi-Yau structure, Abouzaid's split-generation criterion, and their analogues when weak bounding cochains are included. We then turn to computations of the Fukaya category of the hypersurface $X$: we construct a configuration of monotone Lagrangian spheres in $X$, and compute the associated disc potential. The result coincides with the Hori-Vafa superpotential for the mirror of $X$ (up to a constant shift in the Fano index $1$ case). As a consequence, we give a proof of Kontsevich's homological mirror symmetry conjecture for $X$. We also explain how to extract non-trivial information about Gromov-Witten invariants of $X$ from its Fukaya category.

Localized Standard Versus Reduced Formula and Genus 1 Local Gromov–Witten Invariants

For local Calabi–Yau (CY) manifolds which are total spaces of vector bundle over algebraic Goresky-Kottwitz-MacPherson (GKM) manifolds, we propose a formal definition of reduced genus 1 Gromov–Witten (GW) invariants, by assigning contributions from the refined decorated rooted trees. We show that this definition satisfies a localized version of the standard versus reduced (LSvR) formula, whose global version in the compact cases is due to Zinger. As an application, we prove the conjecture in a previous article on the genus 1 GW invariants of local CY manifolds which are total spaces of concave splitting vector bundles over projective spaces. In particular, we prove the mirror formulae for genus 1 GW invariants of |$K_{\mathbb {P}^{2}}$| and |$K_{\mathbb {P}^{3}}$|⁠, conjectured by Klemm, Zaslow, and Pandharipande. In the appendix, we derive the modularity of genus 1 GW invariants for the local |$\mathbb {P}^{2}$| as a consequence of the results on Ramanujan's cubic transformation. Inspired by the LSvR formula, we show that the ordinary genus 1 GW invariants of CY hypersurfaces in projective spaces can be computed by virtual localization and quantum hyperplane property after the contribution of a genus 1 vertex is replaced by a modified one.